Stationary machinery within automated, industrial manufacturing and fabrication processes typically operates in very repetitive, measured, predictable cycles. In these environments, total lost energy over the course of a given time period, say, a day, can be measured or calculated rather easily due to regular, predictable motion cycles.

When it comes to mobile equipment, the cycles are not as regular or predictable, even though they operate “sort of repetitively.” Significant variations occur in each cycle, making lost-power calculation somewhat more challenging.

Vexing variables

Consider the operation of a backhoe excavating a small area, such as a graveyard plot. Variables quickly emerge even in such a confined dig, such as the depth as the project progresses; the reach; and the lift and swing as the dirt piles up. In addition, the operator may not handle the controls in exactly the same manner from cycle to cycle, or the operator may change on another similar project. The number of functions that an operator can manage simultaneously is limited to some degree by the manual dexterity of the operator.

Further variations include soil conditions as the vehicle moves from site to site, and the amount of moisture in the soil, which affects the weight of a bucketful as well as digging resistance. There are doubtless others, but the point is that often widely varying operating conditions make the excavator manufacturer’s job of tuning a machine to “optimal” conditions much more difficult.

The discussion here, though, is hydrostatic transmissions, not excavators. Envisage one used in the drive train of a front-end loader. Its cycle also varies over the course of a day for reasons similar to those already mentioned, as well as others that are particular to front-end loaders. One such example is the amount of travel distance between the load pickup point and the load dump point.

The point is that regular, predictable parts — along with random variations — exist within cyclic operation. Handling these random effects requires statistical methods. For hydrostatic transmission (HSTs), the approach is to use a histogram of the “probability of finding a speed versus the speed” itself, Figure 1.

Generating the histogram

This discussion is not intended to be a treatise on the generalities of statistics or the theory of histograms and their development. Nonetheless, for the statistically challenged reader, it’s helpful to review the steps involved in histogram creation.

First, it’s important to note that the histograms contained herein are not the result of actual measurements on mobile machinery. Instead, they were developed totally through what the Germans call a gedankenexperiment, that is, by theoretical (mental) estimates. However, the following discussion should explain that the values presented are reasonable for a wheel-driven front-end loader and the histogram creation process as described. The lack of actual histogram data does not in any way affect the procedure or the manner in which conclusions are reached.

To start the explanation, one must interpret the data provided in Figure 1. It consists of a series of 35 bars, each being 100 rpm wide. The vertical axis is simply “Number of occurrences.” A total of 73,980 sampleswent into the chart. The sample counts were put into a series of 100-rpm-wide data buckets; each data bucket is a bar on the graph.

Referring to Figure 1, the height of the bar (a data bucket) with a peak value that’s centered at 2850 rpm, which covers the interval from 2800 to 2900 rpm, represents the number of samples found in that interval. That is, of the 73,980 rpm samples — about 4460 of them — lay between the 2800 and 2900 data bucket. Or, in other words, of the 73,980 speed samples, 4460 of them were in the interval from 2800 to 2900, where the notation 2900 means any tiny amount less than 2900 rpm.

To be useful, the histogram must be converted into one that indicates the “probability of occurring,” expressed as a percentage. This task simply requires dividing each bar amplitude by the total number of samples. In the 2800 to 2900 interval, with its 4460 samples, the result is 4460 ÷ 73,980, about 6.03%.

Thus, the modified histogram turns into a probability curve, Figure 2. Therefore, if the machine is operated in a statistically similar manner to that used to generate the histogram data, it will run between 2800 and 2900 rpm about 6% of the time. It will also run, say, between 900 and 1000 rpm about 3.5% of the time.